Orbital symmetry-adapted

S-S Sip Sop SALC SCF SO SOMO T tds tdt tfd TP UV xo 9m 9l 9 P Dithiolene or dithiolate chelate In-plane dithiolene S p orbital Out-of-plane dithiolene S p orbital Symmetry adapted linear combination Self-consistent held Sulfite oxidase Singly occupied molecular orbital Tesla (Trihuoromethyl)ethylenediselenato Toluene-2,3-dithiolate Bis(trihuoromethyl)-1,2-dithiete Trigonal prismatic Ultraviolet Xanthine oxidase Metal-based function Ligand-based function Symmetric in-plane dithiolene molecular orbital... 

Fig. 33a-c. d-valence electron distribution at the (111) surface, a. The out-of-plane lobes of the degenerate d,y, dy, and d atomic orbitals [37]. b. The linear combinations of the plane lobes of the djy, dy, and d, atomic orbitals symmetry adapted to the (111) surface, c. Scheme of surface d-electron density of states... 

In this appendix are presented three different methods of obtaining the linear combinations of Table 6.2, repeated as Table A6.1, the ligand a orbital symmetry-adapted linear combinations for an octahedral complex. There is much to be learnt from a careful comparison of all three methods. 

It is recommended that the reader become familiar with the point-group symmetry tools developed in Appendix E before proceeding with this section. In particular, it is important to know how to label atomic orbitals as well as the various hybrids that can be formed from them according to the irreducible representations of the molecule s point group and how to construct symmetry adapted combinations of atomic, hybrid, and molecular orbitals using projection operator methods. If additional material on group theory is needed. Cotton s book on this subject is very good and provides many excellent chemical applications. 

Because symmetry operators eommute with the eleetronie Hamiltonian, the wavefunetions that are eigenstates of H ean be labeled by the symmetry of the point group of the moleeule (i.e., those operators that leave H invariant). It is for this reason that one eonstruets symmetry-adapted atomie basis orbitals to use in forming moleeular orbitals. 

For example, the three NH bonding and three NH antibonding orbitals in NH3, when symmetry adapted within the C3V point group, cluster into ai and e mos as shown in the Figure below. The N-atom localized non-bonding lone pair orbital and the N-atom Is core orbital also belong to ai symmetry. 

Define the symmetry adapted "eore" and "valenee" orbitals of the following systems ... 

An example will help illustrate these ideas. Consider the formaldehyde molecule H2CO in C2v symmetry. The configuration which dominates the ground-state waveflinction has doubly occupied O and C 1 s orbitals, two CH bonds, a CO a bond, a CO n bond, and two 0-centered lone pairs this configuration is described in terms of symmetry adapted orbitals as follows (Iai22ai23ai2lb2 ... 

The spin- and spatial- symmetry adapted N-eleetron funetions referred to as CSFs ean be formed from one or more Slater determinants. For example, to deseribe the singlet CSF eorresponding to the elosed-shell orbital oeeupaney, a single Slater determinant... 

The single Slater determinant wavefunction (properly spin and symmetry adapted) is the starting point of the most common mean field potential. It is also the origin of the molecular orbital concept. 

To illustrate sueh symmetry adaptation, eonsider symmetry adapting the 2s orbital of N and the three Is orbitals of H. We begin by determining how these orbitals transform under the symmetry operations of the C3V point group. The aet of eaeh of the six symmetry operations on the four atomie orbitals ean be denoted as follows ... 

IV. Projeetor Operators Symmetry Adapted Einear Combinations of Atomie Orbitals... 

To generate the above Ai and E symmetry-adapted orbitals, we make use of so-ealled symmetry projeetion operators Pe and Pa These operators are given in terms of... 

These equations state that the three Ish orbitals can be combined to give one Ai orbital and, since E is degenerate, one pair of E orbitals, as established above. With knowledge of the ni, the symmetry-adapted orbitals can be formed by allowing the projectors... 

We now return to the symmetry analysis of orbital produets. Sueh knowledge is important beeause one is routinely faeed with eonstrueting symmetry-adapted N-eleetron eonfigurations that eonsist of produets of N individual orbitals. A point-group symmetry operator S, when aeting on sueh a produet of orbitals, gives the produet of S aeting on eaeh of the individual orbitals... 

In order to obtain this savings in the computational cost, orbitals are symmetry-adapted. As various positive and negative combinations of orbitals are used, there are a number of ways to break down the total wave function. These various orbital functions will obey different sets of symmetry constraints, such as having positive or negative values across a mirror plane of the molecule. These various symmetry sets are called irreducible representations. 

Molecular orbitals are not unique. The same exact wave function could be expressed an infinite number of ways with different, but equivalent orbitals. Two commonly used sets of orbitals are localized orbitals and symmetry-adapted orbitals (also called canonical orbitals). Localized orbitals are sometimes used because they look very much like a chemist s qualitative models of molecular bonds, lone-pair electrons, core electrons, and the like. Symmetry-adapted orbitals are more commonly used because they allow the calculation to be executed much more quickly for high-symmetry molecules. Localized orbitals can give the fastest calculations for very large molecules without symmetry due to many long-distance interactions becoming negligible. 

There is always a transformation between symmetry-adapted and localized orbitals that can be quite complex. A simple example would be for the bonding orbitals of the water molecule. As shown in Figure 14.1, localized orbitals can... 

Most ah initio calculations use symmetry-adapted molecular orbitals. Under this scheme, the Hamiltonian matrix is block diagonal. This means that every molecular orbital will have the symmetry properties of one of the irreducible representations of the point group. No orbitals will be described by mixing dilferent irreducible representations. 

ADF uses a STO basis set along with STO fit functions to improve the efficiency of calculating multicenter integrals. It uses a fragment orbital approach. This is, in essence, a set of localized orbitals that have been symmetry-adapted. This approach is designed to make it possible to analyze molecular properties in terms of functional groups. Frozen core calculations can also be performed. 

An orbital correlation diagram can be constructed by examining the symmetry of the reactant and product orbitals with respect to this plane. The orbitals are classified by symmetry with respect to this plane in Fig. 11.9. For the reactants ethylene and butadiene, the classifications are the same as for the consideration of electrocyclic reactions on p. 610. An additional feature must be taken into account in the case of cyclohexene. The cyclohexene orbitals tr, t72. < i> and are called symmetry-adapted orbitals. We might be inclined to think of the a and a orbitals as localized between specific pairs of carbon... 

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